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1.
Verification of the unitary similarity between matrices having quadratic minimal polynomials is known to be much cheaper than the use of the general Specht-Pearcy criterion. Such an economy is possible due to the following fact: n × n matrices A and B with quadratic minimal polynomials are unitarily similar if and only if A and B have the same eigenvalues and the same singular values. It is shown that this fact is also valid for a subclass of matrices with cubic minimal polynomials, namely, quadratically normal matrices of type 1.  相似文献   

2.
Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.  相似文献   

3.
A matrix A ∈ Mn(C) is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.  相似文献   

4.
Normal matrices in which all principal submatrices are normal are said to be principally normal. Various characterizations of irreducible matrices in this class of are given. Notably, it is shown that an irreducible matrix is principally normal if and only if it is normal and all of its eigenvalues lie on a line in the complex plane. Such matrices provide a generalization of the Cauchy interlacing theorem.  相似文献   

5.
We study various conditions on matricesB andC under which they can be the off-diagonal blocks of a partitioned normal matrix. To Kalyan Sinha on his sixtieth birthday  相似文献   

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An almost normal matrix is defined as an n by n matrix having n − 1 mutually orthogonal eigenvectors. The properties of these matrices are shown to be intermediate between the properties of conventional normal matrices and those of general matrices. In particular, the Schur form of an almost normal matrix can in a certain sense be considered canonical.  相似文献   

9.
In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.  相似文献   

10.
Let A be a normal n×n matrix. This paper discusses in detail under what conditions and in what way A can be dilated to a normal matrix of order n+1 or n+2. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 63–94. Translated by Kh. D. Ikramov.  相似文献   

11.
Augmented nodal matrices play an important role in the analysis of different features of electrical circuit models. Their study can be addressed in an abstract setting involving two- and three-colour weighted digraphs. By means of a detailed characterization of the structure of proper and normal trees, we provide a unifying framework for the rank analysis of augmented matrices. This covers in particular Maxwell’s tree-based determinantal expansions of (non-augmented) nodal matrices, which can be considered as a one-colour version of our results. Via different colour assignments to circuit devices, we tackle the DC-solvability problem and the index characterization of certain differential-algebraic models which arise in the nodal analysis of electrical circuits, extending several known results of passive circuits to the non-passive context.  相似文献   

12.
The Hermitian Lanczos method for Hermitian matrices has a well-known connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of . This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomials to the set of polyanalytic polynomials. Denoting by the number of elements computed so far, the arising scheme yields a recurrence of length bounded by for polyanalytic polynomials orthogonal on a discrete subset of . Like this slowly growing length of the recurrence, the method preserves, at least partially, the properties of the Hermitian Lanczos method. We employ the algorithm in least squares approximation and bivariate Lagrange interpolation.

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13.
Given a normal matrix A, asymptotic bounds are obtained for |Am| in terms of the spectral radius of A, the number of eigenvalues of A with modulus equal to the spectral radius of A, and the order of A. These results are extended to provide bounds for |Am| for all m ? 1.  相似文献   

14.
When can an (n-k)×(n-k) normal matrix B be imbedded in an n×n normal matrix A? This question was studied for the first time 50 years ago by Ky Fan and Gordon Pall, who gave the complete answer in the case k=1. Since then, a few authors obtained additional results. In this note, we show how an approach inspired by the Hermitian case can throw some light on the problem.  相似文献   

15.
To any complex function there corresponds a Fourier series, which is often associated with a sequence {T n} of Toeplitz n × n matrices. Functions whose Fourier series generate sequences of normal Toeplitz matrices are classified, and a procedure for constructing Fourier series for which the sequence {T n} contains an infinite subsequence of normal matrices is described.  相似文献   

16.
We study a class of weighted shifts W α defined by a recursively generated sequence α ≡ α0, … , α m−2, (α m−1, α m , α m+1) and characterize the difference between quadratic hyponormality and positive quadratic hyponormality. We show that a shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisfies a finite number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, W α is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for quadratic hyponormality of recursively generated weighted shift W α (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift W α need not be positively quadratically hyponormal.  相似文献   

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In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset S. It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group G is normal iff G is either abelian or has the form for some non-negative integer n, where Q8 is the quaternion group and is the abelian group of order 2n and exponent 2.  相似文献   

19.
Summary The Householder-Givens method for the solution of the Hermitean eigenproblem is used for the solution of the skew-Hermitean and the normal eigenproblem.This research was performed at the (Mathematical Institute of the) State University in Utrecht.  相似文献   

20.
In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought.  相似文献   

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